28 PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
Table No. 89 (continued). 
(8, 12) 
AB 3 D-B 3 C + 2BCH -C 2 G + P=0 
-3ADH -2BCII + 2C 2 G + 18CD 3 + FK-2P=0 
EL + FK— 2I 2 =0 
(8, 10) 
ABK- CN-6DL-2FJ+ HI = 0 
AP +2CN+ FJ = 0 
B-I - CN+3DL+ FJ-2HI = 0 
(8, 8) 
ABJ — B 4 +4B 2 H— 9BD 2 + 12CM-EK-3H 2 =0 
ADG +2B 2 H-12BD 2 + 8CM-EK-2H 2 =0 
(8, 6) 
AO + 6DK-3EJ+2GI = 0 
BN + 3DK- EJ+ GI = 0 
342. In illustration take any one of the lines of Table No. 88, for instance the line 
(7, 17) | 4 | A 2 BE, A 2 L, ACI, ADF, BCF, C 2 E | 2 f | 
there are here 6 composite covariants, but the number of asyzygetic covariants is = 4 ; 
there must therefore be 6 — 4, — 2 syzygies ; we have however (see Table No. 89) two 
derived syzygies of the right form, viz. these are 
A(AL-2CI+3DF)=0, 
C(AI+ BF- CE)=0, 
which are designated as 2%', and there is consequently no new syzygy 2. 
But in the line 
(7, 15) | 5 | A 3 G, A 2 BE>, AB 2 C, ACH, AE 2 , C 2 D, FI | 2', 2 | 
there are 7 composite covariants, but the number of asyzygetic covariants is = 5 ; there 
must therefore be 7 — 5, =2 syzygies. One of these is the derived syzygy 
A( A 2 G - E 2 — 1 2 ABD - 4B 2 C) = 0, 
which is designated by 2' ; the other is a new syzygy (see Table No. 89), 
A 2 BD - ABC 2 + ACH - 6 C 2 D - FI = 0, 
designated by %. 
343. Take now the line 
(8, 20) | 6 | A 4 G, A 3 BD, A 2 B 2 C, A 2 CH, A 2 E 2 , AC 2 D, AFI, BC 3 , BF 2 , CEF | 52', <r | ; 
there are here 10 composite covariants, but the number of irreducible covariants is =6 ; 
there should therefore be 10 — 6, =4 syzygies. There are, however, the 5 derived syzygies 
A 2 ( A 2 G — 1 2 ABD — 4B 2 C — E 2 ) = 0, &c. (see Table No. 89) 
designated by 52' ; since these are equivalent to 4 syzygies only there must be 1 identical 
relation between them (designated by o), viz. this is the equation 0 = 0 obtained by 
adding the several syzygies, multiplied each by the proper numerical factor as shown 
Table No. 89. 
